Consider the set of eight vectors V = a i ^ + b j ^ + c k ^ : a, b, c  ∈ - 1 , 1 . Three…

Consider the set of eight vectors  V = a i ^ + b j ^ + c k ^ : a, b, c  ∈ - 1 , 1 . Three non-coplanar vectors can be chosen from V in 2 p ways. Then p is

Solution


x → ≡ i ^ + j ^ + k ^
y → ≡ i ^ + j ^ - k ^
z → ≡ i ^ + j ^ + k ^
ω → = i ^ - j ^ - k ^
x ¯ ′ = - i ^ - j ^ - k ^
y → ′ = - i ^ - j ^ + k ^
z → ′ = - i ^ + j ^ - k ^
ω → ′ = - i ^ + j ^ + k ^
If we take x → ,  x → ′  and any one of remaining six, vectors will always be coplanar
∴     No. of coplanar vectors = 6
Similarly on taking y → , y → ′ = 6
z , z → ′ = 6
ω , ω → ′ = 6
∴    No. of set coplanar vectors = 24
8 C 3 - 2 4 = 3 2

Asked in: JEE Advanced 2013 (Paper 1)

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